On c-lineability

Abstract

In the paper we study $c$-lineability and $c$-spaceability of some families $F$ of real functions defined on an interval I. The main goal is to formulate general conditions under which any non-empty family $F\subset R^I$ of functions is $c$-spaceable or $c$-lineable. Generally, we consider the families of function of the form $F=F_1\setminus F_2$. In most cases, families of functions for which lineability and spaceability are studied have such a form. Most often, family $F_2$ is seemingly “very close” to $F_1$ or consists of “almost all” functions. The results obtained in this paper are a generalization of previous ideas. The main idea of our constructions is to “reproduce” one function to obtain $c$-dimensional (closed) linear space. For this “reproduction” we use the Fichtenholz-Kantorovich Theorem, applied to a countable family of pairwise disjoint intervals contained in the domain of functions from the considered class. The initial function is “squashed” and “pasted” into disjoint intervals included in the domain of constructed function.